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      Class 11 MATHS – JEE

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      • Class 11
      • Class 11 MATHS – JEE
      CoursesClass 11MathsClass 11 MATHS – JEE
      • 1.Sets, Relation and Functions
        12
        • Lecture1.1
          Introduction to sets, Description of sets 32 min
        • Lecture1.2
          Types of Sets, Subsets 39 min
        • Lecture1.3
          Intervals, Venn Diagrams, Operations on Sets 37 min
        • Lecture1.4
          Laws of Algebra of Sets 26 min
        • Lecture1.5
          Introduction to sets and its types, operations of sets, Venn Diagrams 28 min
        • Lecture1.6
          Functions and its Types 38 min
        • Lecture1.7
          Functions Types 17 min
        • Lecture1.8
          Cartesian Product of Sets, Relation, Domain and Range 40 min
        • Lecture1.9
          Sum Related to Relations 04 min
        • Lecture1.10
          Sums Related to Relations, Domain and Range 22 min
        • Lecture1.11
          Chapter Notes – Sets, Relation and Functions
        • Lecture1.12
          NCERT Solutions – Sets, Relation and Functions
      • 2.Trigonometric Functions
        28
        • Lecture2.1
          Introduction, Some Identities and Some Sums 16 min
        • Lecture2.2
          Some Sums Related to Trigonometry Identities, trigonometry Functions Table and Its Quadrants 35 min
        • Lecture2.3
          NCERT Sums Ex.3.3 (Q.1-5)Based on Trigometry table and Their Quadrants, Trigonometry Identities of Sum and Diff. of two Angles 21 min
        • Lecture2.4
          NCERT Sums Ex-3.2 Based on Trigonometry Function of Lower & Higher Angles 22 min
        • Lecture2.5
          NCERT Sums Ex-3.3 (Q.6 – 10) Based on Radian Angles 11 min
        • Lecture2.6
          NCERT Sums Ex-3.3 (Q.11-13)Based on Trigonometry Identities 16 min
        • Lecture2.7
          NCERT Sums Ex-3.3 (Q. 14)Based on Trigonometry Identities 14 min
        • Lecture2.8
          NCERT Sums Ex-3.3 (Q.16) Based on Trigonometry Identities 05 min
        • Lecture2.9
          NCERT Sums Ex-3.3 (Q.17 -21) Based on Trigonometry Identities 12 min
        • Lecture2.10
          NCERT Sums Ex-3.4 (Q. 1 – 9), Trigonometry Equation 25 min
        • Lecture2.11
          Sums Based on Trigonometry Equations 24 min
        • Lecture2.12
          Sums Based on Trigonometry Equations 11 min
        • Lecture2.13
          Sums Based on Trigonometry Equations 11 min
        • Lecture2.14
          Sums Based on Trigonometry Equations 17 min
        • Lecture2.15
          Equations Having two Variable Angle which satisfy both equations 10 min
        • Lecture2.16
          Trigonometrical Identities-Some important relations and Its related Sums 16 min
        • Lecture2.17
          Sums Related to Trigonometrical Identities 18 min
        • Lecture2.18
          Properties of Triangles and Solution of Triangles-Sine formula, Napier Analogy and Sums 17 min
        • Lecture2.19
          Relation Between Degree and Radian, Quadrant and NCERT Sum Ex.3.1, 3.2 41 min
        • Lecture2.20
          Trigonometric Functions Table 09 min
        • Lecture2.21
          Some Trigonometric Identities and its related Sums 42 min
        • Lecture2.22
          Sums Related to Trigonometrical Identities 19 min
        • Lecture2.23
          Sums Related to Trigonometrical Identities 41 min
        • Lecture2.24
          Sums Related to Trigonometrical Identities 23 min
        • Lecture2.25
          Trigonometry Equations 44 min
        • Lecture2.26
          Sum Based on Trigonometry Equations 07 min
        • Lecture2.27
          Sums Based on Trigonometry function of Lower Angle 03 min
        • Lecture2.28
          Chapter Notes – Trigonometric Functions
      • 3.Mathematical Induction
        5
        • Lecture3.1
          Introduction to PMI 25 min
        • Lecture3.2
          NCERT Solution of EX- 4.1 14 min
        • Lecture3.3
          NCERT Solution of EX- 4.1 22 min
        • Lecture3.4
          NCERT Solution of EX- 4.1 13 min
        • Lecture3.5
          Chapter Notes – Mathematical Induction
      • 4.Complex Numbers and Quadratic Equation
        15
        • Lecture4.1
          Introduction, Nature of Roots, Numbers, Introduction of i 27 min
        • Lecture4.2
          Sum Related to Relations, Real and Imaginary part of C-N, Conjugate of a C-N 33 min
        • Lecture4.3
          Absolute value or Modulus of a C-N and Related Sums 29 min
        • Lecture4.4
          Sums Related To Multiplicative Inverse 29 min
        • Lecture4.5
          Polar Form of a C-N 32 min
        • Lecture4.6
          Sums Related To Polar Form 32 min
        • Lecture4.7
          Square Roots of C-N and its Related Sums 28 min
        • Lecture4.8
          De Moivris Theorem and its related Sums 31 min
        • Lecture4.9
          Introduction, Nature of Roots, Numbers, Introduction of i and its Sums, Real and Imaginary Part of C-N 35 min
        • Lecture4.10
          Sums Related to Real and Imaginary Part of C-N and Operations on C-N 13 min
        • Lecture4.11
          Sums Related To Multiplicative Inverse 07 min
        • Lecture4.12
          Sums Related To Multiplicative Inverse and Modulus and Argument of a C-N 33 min
        • Lecture4.13
          Polar form of a C-N, Nature of Roots 38 min
        • Lecture4.14
          Sums Based on Roots of Quadratic Equations, Sums of Polar form 10 min
        • Lecture4.15
          Chapter Notes – Complex Numbers and Quadratic Equation
      • 5.Linear Inequalities
        4
        • Lecture5.1
          Introduction, Solve some Linear Inequalities and its Graph 42 min
        • Lecture5.2
          Solve some Linear Inequalities and its Graph 12 min
        • Lecture5.3
          Solve some Linear Inequalities and its Graph and Introduction-Permutations and Combinations 35 min
        • Lecture5.4
          Chapter Notes – Linear Inequalities
      • 6.Permutations and Combinations
        5
        • Lecture6.1
          NCERT Sums Ex-7.3, Equation 41 min
        • Lecture6.2
          NCERT Sums Ex-7.3, Equation 02 min
        • Lecture6.3
          Combination and NCERT Sums Ex-7.4 40 min
        • Lecture6.4
          NCERT Sums Ex-7.1 & 7.2 22 min
        • Lecture6.5
          Chapter Notes – Permutations and Combinations
      • 7.Binomial Theorem
        19
        • Lecture7.1
          Introduction to Binomial Theorem 21 min
        • Lecture7.2
          Binomial General Expansion and Their Derivations and its Related Sums 22 min
        • Lecture7.3
          Pascal’s Triangle Theorem, Addition of Two Expansion, NCERT Sums Ex-8.1 26 min
        • Lecture7.4
          Sums of Miscellaneous Exercise and Ex-8.1, Finding the Any Term from nth Term 42 min
        • Lecture7.5
          NCERT Sums Ex-8.1 14 min
        • Lecture7.6
          NCERT Sums Ex-8.1 04 min
        • Lecture7.7
          NCERT Sums Ex-8.2, Middle Term 21 min
        • Lecture7.8
          NCERT Sums Ex-8.2, Middle Term Related Sums 08 min
        • Lecture7.9
          To Find the Coefficient of X^r in the Expansion of (X+A)^n, NCERT Sums Ex-8.2 and Miscellaneous Ex. 40 min
        • Lecture7.10
          NCERT Sums Ex-8.2 10 min
        • Lecture7.11
          To Find the Sum of the Coefficients in the Expansion of (1+x)^n and its Related Sums 27 min
        • Lecture7.12
          Sums Related to Binomials Coefficients 24 min
        • Lecture7.13
          Binomial Theorem for any Index and its Related Sums 27 min
        • Lecture7.14
          Introduction to Binomial Theorem, General Term in the Expansion of (x+a)^n. 39 min
        • Lecture7.15
          NCERT Sums Ex-8.1 & 8.2, Pascals’ Triangle, pth Term from End 24 min
        • Lecture7.16
          Sums related to Finding the Coefficient, NCERT Sums Ex-8.2, Middle Term 40 min
        • Lecture7.17
          Sums Related to Middle Term 17 min
        • Lecture7.18
          Sums Related to Coefficient of the Any Term 31 min
        • Lecture7.19
          Chapter Notes – Binomial Theorem
      • 8.Sequences and Series
        14
        • Lecture8.1
          Introduction, A.P., nth Term and Sum of nth Term, P Arithmetic Mean B/w a and b, Sum Based on Fibonacci Sequence 27 min
        • Lecture8.2
          NCERT Sums Ex-9.2 37 min
        • Lecture8.3
          NCERT Sums Ex-9.2 18 min
        • Lecture8.4
          NCERT Sums Ex-9.2, Geometric Progression -Introduction, nth term, NCERT Sums Ex-9.3 39 min
        • Lecture8.5
          NCERT Sums Ex-9.3 16 min
        • Lecture8.6
          Sum of n term of G.P., NCERT Sums Ex-9.3 40 min
        • Lecture8.7
          NCERT Sums Ex-9.3 08 min
        • Lecture8.8
          NCERT Sums Ex-9.3, Insert P Geometrical Mean B/w a and b 36 min
        • Lecture8.9
          NCERT Sum Ex-9.3 17 min
        • Lecture8.10
          NCERT Sum Ex-9.3 09 min
        • Lecture8.11
          Some Special Series, NCERT Sum Ex-9.4 36 min
        • Lecture8.12
          NCERT Sum Ex-9.4 02 min
        • Lecture8.13
          NCERT Sum Ex-9.4 18 min
        • Lecture8.14
          Chapter Notes – Sequences and Series
      • 9.Properties of Triangles
        2
        • Lecture9.1
          Sine 7 Cosine Rule, Projection Formulae, Napier’s Analogy, Incircle, Some Sums 41 min
        • Lecture9.2
          Angle of Elevations and Depression. and Its Related Sums 13 min
      • 10.Straight Lines
        30
        • Lecture10.1
          Introduction, Equation of Line, Slope or Gradient of a line 24 min
        • Lecture10.2
          Sums Related to Finding the Slope, Angle Between two Lines 22 min
        • Lecture10.3
          Cases for Angle B/w two Lines, Different forms of Line Equation 23 min
        • Lecture10.4
          Sums Related Finding the Equation of Line 27 min
        • Lecture10.5
          Sums based on Previous Concepts of Straight line 32 min
        • Lecture10.6
          Parametric Form of a Straight Line 16 min
        • Lecture10.7
          Sums Related to Parametric Form of a Straight Line 16 min
        • Lecture10.8
          Sums Based on Concurrent of lines, Angle b/w Two Lines 45 min
        • Lecture10.9
          Different condition for Angle b/w two lines 04 min
        • Lecture10.10
          Sums Based on Angle b/w Two Lines 36 min
        • Lecture10.11
          Equation of Straight line Passes Through a Point and Make an Angle with Another Line 09 min
        • Lecture10.12
          Sums Based on Equation of Straight line Passes Through a Point and Make an Angle with Another Line 15 min
        • Lecture10.13
          Sums Based on Equation of Straight line Passes Through a Point and Make an Angle with Another Line 17 min
        • Lecture10.14
          Finding the Distance of a point from the line 34 min
        • Lecture10.15
          Sum Based on Finding the Distance of a point from the line and B/w Two Parallel Lines 33 min
        • Lecture10.16
          Sums Based on Find the Equation of Bisector of Angle Between two intersecting Lines 44 min
        • Lecture10.17
          Sums Based on Find the Equation of Bisector of Angle Between two intersecting Lines 02 min
        • Lecture10.18
          Introduction, Distance B/w Two Points, Slope, Equation of Line 32 min
        • Lecture10.19
          NCERT Sums Ex-10.1 43 min
        • Lecture10.20
          NCERT Sums Ex-10.1 29 min
        • Lecture10.21
          NCERT Sums Ex-10.1 & 10.2 43 min
        • Lecture10.22
          NCERT Sums Ex-10.2 30 min
        • Lecture10.23
          NCERT Sums Ex-10.2 41 min
        • Lecture10.24
          NCERT Sums Ex-10.2 & 10.3 21 min
        • Lecture10.25
          NCERT Sums Ex- 10.3 (Reduce the Equation into intercept Form, Normal form) 42 min
        • Lecture10.26
          NCERT Sums Ex-10.3 21 min
        • Lecture10.27
          NCERT Sums Ex-10.3 (Equation of Parallel line, Perpendicular Line of given line, Sums Based of Angle B/w Two Lines) 42 min
        • Lecture10.28
          NCERT Sums Ex-10.3 09 min
        • Lecture10.29
          NCERT Sums Ex-10.3 26 min
        • Lecture10.30
          Chapter Notes – Straight Lines
      • 11.Conic Sections
        21
        • Lecture11.1
          Introduction, General Equation of second Degree, Parabola, Sums based on Finding Equation of Parabola 41 min
        • Lecture11.2
          Sums Based on Equation of Parabola, Four Forms of Parabola-Form (i) 30 min
        • Lecture11.3
          Sums Based on Four Forms of Parabola-Form (i) 32 min
        • Lecture11.4
          Four Forms of Parabola-Form (ii), (iii) (iv) 13 min
        • Lecture11.5
          Sums Based on Four forms of Parabola 18 min
        • Lecture11.6
          Position of a Point with Respect to Parabola and its Sums 43 min
        • Lecture11.7
          Circles-Introduction, Different Cases for Circle Equations, NCERT Sums Ex-11.1 16 min
        • Lecture11.8
          NCERT Sums Ex-11.1 40 min
        • Lecture11.9
          Circle Important Point Revise, Intersection of Axes, NCERT Sums Ex-11.1 11 min
        • Lecture11.10
          NCERT Sums Ex-11.1 44 min
        • Lecture11.11
          Parabola- Introduction, General Equation , Sums, Some Important Concepts for Parabola 12 min
        • Lecture11.12
          Different Form of Parabola, NCERT Sum Ex-11.2 13 min
        • Lecture11.13
          NCERT Sum Ex-11.2 34 min
        • Lecture11.14
          Ellipse-Introduction, General Equation, NCERT Sums Ex-11.3 36 min
        • Lecture11.15
          NCERT Sums Ex-11.3 02 min
        • Lecture11.16
          NCERT Sums Ex-11.3 23 min
        • Lecture11.17
          Hyperbola-Introduction, NCERT Sums Ex-11.4 12 min
        • Lecture11.18
          NCERT Sums Ex-11.4 25 min
        • Lecture11.19
          Chapter Notes – Conic Sections Circles
        • Lecture11.20
          Chapter Notes – Conic Sections Ellipse
        • Lecture11.21
          Chapter Notes – Conic Sections Parabola
      • 12.Coordinate Geometry
        8
        • Lecture12.1
          Introduction to Rectangular Cartesian Coordinate Geometry (2D), Distance b/w two points 23 min
        • Lecture12.2
          Cartesian Coordinate of points 32 min
        • Lecture12.3
          Questions rel to cartesian coordinate of points 25 min
        • Lecture12.4
          Section Formula – Case 1, Case 2 24 min
        • Lecture12.5
          Problem Solving 26 min
        • Lecture12.6
          Centeroid, Incenter, Circumcenter of a triangle 30 min
        • Lecture12.7
          Locus Problems 17 min
        • Lecture12.8
          Problem Solving 21 min
      • 13.Three Dimensional Geometry
        3
        • Lecture13.1
          Introduction to 3D 18 min
        • Lecture13.2
          Numerical problems 14 min
        • Lecture13.3
          Chapter Notes – Three Dimensional Geometry
      • 14.Limits And Derivatives
        12
        • Lecture14.1
          Introduction to limits 42 min
        • Lecture14.2
          EX-13.1 16 min
        • Lecture14.3
          Questions based on algebra of limits 41 min
        • Lecture14.4
          Limits of a polynomial 12 min
        • Lecture14.5
          rational function 37 min
        • Lecture14.6
          trigo function 21 min
        • Lecture14.7
          Introduction to Derivatives 37 min
        • Lecture14.8
          Ex-13.2 22 min
        • Lecture14.9
          Algebra of derivatives 38 min
        • Lecture14.10
          Derivative of polynomial 13 min
        • Lecture14.11
          trigo function 11 min
        • Lecture14.12
          Chapter Notes – Limits And Derivatives
      • 15.Mathematical Reasoning
        3
        • Lecture15.1
          What is statement ? Special word and phrases, negation of statement , Compound statement , and & or in compound statement , truth table Solving the problems of Ex- 14.1 , 14.2 25 min
        • Lecture15.2
          Solving Ex-14.3, Ex-14.4, Implications, Validating statements, Ex-14.5, Direct method 24 min
        • Lecture15.3
          Chapter Notes – Mathematical Reasoning
      • 16.Statistics
        5
        • Lecture16.1
          Mean, Median, Mode, Range, Mean Deviation Solution of Ex-15.1 27 min
        • Lecture16.2
          Mean Deviation about Mean & Median, Ex-15.2, Mean and Variance, Standard deviation 35 min
        • Lecture16.3
          Ex-15.2 , Variance and Standard deviation 09 min
        • Lecture16.4
          Ex-15.3, Analysis of frequency distribution, comparison of two frequency distribution with same mean 23 min
        • Lecture16.5
          Chapter Notes – Statistics
      • 17.Probability
        3
        • Lecture17.1
          Outcomes & sample space, Ex. 16.3 19 min
        • Lecture17.2
          Ex.16.3, Probability of an event, Algebra of event 38 min
        • Lecture17.3
          Chapter Notes – Probability
      • 18.Binary Number
        2
        • Lecture18.1
          Binary numbers, Conversion of Binary to Decimal and Decimal to binary 45 min
        • Lecture18.2
          Addition, Subtraction, Multiplication, Division 02 min

        Chapter Notes – Conic Sections Parabola

        A hyperbola is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point in the same plane to its distance from a fixed line is always constant, which is always greater than unity.

        The fixed point is called the focus and the fixed line is directrix and the ratio is the eccentricity.

        Transverse and Conjugate Axes

        The line through the foci of the hyperbola is called its transverse axis.

        The line through the centre and perpendicular to the transverse axis of the hyperbola is called its conjugate axis.

        1. Centre O(0, 0)
        2. Foci are S(ae,0),S1(-ae, 0)
        3. Vertices A(a, 0), A1(-a, 0)
        4. Directrices / : x = a/e, l’ : x = -a/e
        5. Length of latusrectum LL1 = L’L’1 = 2b2/a
        6. Length of transverse axis 2a.
        7. Length of conjugate axis 2b.
        8. Eccentricity  or b2 = a2(e2 – 1)
        9. Distance between foci =2ae
        10. Distance between directrices = 2a/e

         

        Conjugate Hyperbola

        1. (i) Centre O(0, 0)
        2. (ii) Foci are S (0, be), S1(0, — be)
        3. (iii) Vertices A(0, b) , A1(0, — b)
        4. (iv) Directrices

        l:y = b/e, l’ : y = —b/e

        1. (v) Length of latusrectum LL1 = L’ L’1 = 2a2/b
        2. (vi) Length of transverse axis 2b.
        3. (vii) Length of conjugate axis 2a.
        4. (viii) Eccentricity

        1. (ix) Distance between foci = 2be

        10.(x) Distance between directrices = 2b/e

        Focal Distance of a Point

        The distance of a point on the hyperbola from the focus is called it focal distance. The difference of the focal distance of any point on a, hyperbola is constant and is equal to the length of transverse axis the hyperbola i.e.,

        S1P — SP = 2a

        where, S and S1 are the foci and P is any point or P the hyperbola.

        Equation of Hyperbola in Different Form

        1 If the centre of the hyperbola is (h, k) and the directions of the axes are parallel to the coordinate axes, then the equation of the hyperbola, whose transverse and conjugate axes are 2a and 2b is

        2. If a point P(x, y) moves in the plane of two perpendicular straight lines a1x + b1y + c1 = 0 and b1x – a1y + c2 = 0 in such a way that

        Then, the locus of P is hyperbola whose transverse axis lies along b1x – a1y + c2 = 0 and conjugate axis along the line a1x + b1y + c1 = 0. The length of transverse and conjugate axes are 2a and 2b, respectively.

        Parametric Equations

        1. Parametric equations of the hyperbola  x = a sec θ, y = b tan θ

        or x = a cosh θ, y = b sinhθ

        1. The equations  are also the parametric equations of the hyperbola.

         

        Equation of Chord

        1. Equations of chord joining two points P(a sec θ1, b tan θ1,) and Q(a sec θ2, b tan θ2) on the hyperbola

        1. Equations of chord of contact of tangents drawn from a point (x1, y1) to the hyperbola 
        2.  The equation of the chord of the hyperbola bisected at point (x1, y1) is given by

        Equation of Tangent Hyperbola

        1. Point Form The equation of the tangent to the hyperbola 
        2. Parametric Form The equation of the tangent to the

        hyperbola 

        1. Slope Form The equation of the tangents of slope m to

        the hyperbola  The coordinates of the point of contact are

        1. The tangent at the points P(a sec θ1 , b tan θ1) and Q (a sec θ2, b tan θ2) intersect at the point

        1. Two tangents drawn from P are real and distinct, coincident or imaginary according as the roots of the equation m2(h2 – a2) – 2khm + k2 + b2 = 0. are real and distinct, coincident or imaginary.
        2. The line y = mx + c touches the hyperbola, if c2 = a2m2 – b2 the point of contacts 

        Normal Equation of Hyperbola

        1. Point Form The equation of the normal to the hyperbola 
        2. Parametric Form The equation of the normal at (a sec θ, b tan θ) to the hyperbola 

        is ax cos θ + by cot θ = a2 + b2.

        1. Slope Form The equations of the normal of slope m to the hyperbola  are given by

        The coordinates of the point of contact are

        1. The line y = mx + c will be normal to the hyperbola  if,

        1. Maximum four normals can be drawn from a point (x1, y1) to the hyperbola 

         

        Conormal Points

        Points on the hyperbola, the normals at which passes through a given point are called conormal points.

          1. The sum of the eccentric angles of conormal points is an odd ion multiple of π.
          2. If θ1 , θ2 , θ3 and θ4 are eccentric angles of four points on the hyperbola  , then normal at which they are concurrent, then

        (a) ∑cos( θ1 + θ2) = 0

        (b) ∑sin( θ1 + θ2) = 0

          1. If θ1 , θ2 and θ3 are the eccentric angles of three points on the hyperbola  , such that sin(θ1 + θ2) + sin(θ2 + θ3) + sin(θ3 + θ1) = 0. Then, the normals at these points are concurrent.
          2. If the normals at four points P(x1, y1), Q(x2, y2), R(x3 , y3) and S(X4, y4) on the

         

        hyperbola

        are concurrent, then

        Conjugate Points and Conjugate Lines

        1. Two points are said to be conjugate points with respect to a hyperbola, if each lies on the polar of the other.
        2. Two lines are said to be conjugate lines with respect to a hyperbola  , if each passes through the pole of the other.

        Diameter and Conjugate Diameter

        1. Diameter The locus of the mid-points of a system of parallel chords of a hyperbola is called a diameter.>

        The equation of the diameter bisecting a system of parallel chord of slope m to the

         

        hyperbola

        is

        1. Conjugate Diameter The diameters of a hyperbola are sal to be conjugate diameter, if each bisect the chords parallel to th other.

        The diameters y = m1x and y = m2x are conjugate, if m1 m2 = b2/a2.

        1. In a pair of conjugate diameters of a hyperbola, only one mee the hyperbola in real points.

        Asymptote

        An asymptote to a curve is a straight line, at a finite distance from the origin, to which the tangent to a curve tends as the point of contact goes to infinity.

        1. The equation of two asymptotes of the hyperbola  are 
        2. The combined equation of the asymptotes to the hyperbola 
        3. When b = a, i.e., the asymptotes of rectangular hyperbola x2 – y2 = a2 are y = ± x which are at right angle.
        4. A hyperbola and its conjugate hyperbola have the same asymptotes.
        5. The equation of the pair of asymptotes differ the hyperbola and the conjugate hyperbola by the same constant only i.e., Hyperbola — Asymptotes = Asymptotes — Conjugate hyperbola
        6. The asymptotes pass through the centre of the hyperbola.
        7. The bisectors of angle between the asymptotes are the coordinate axes.
        8. The angle between the asymptotes of  is 2 tan-1(b/a) or 2 sec-1(e).

         

        Director Circle

        The locus of the point of intersection of the tangents to the hyperbolo  , which are perpendicular to each other, is called a director circle. The equation of director circle is x2 + y2 = a2 – b2.

        Rectangular Hyperbola

        A hyperbola whose asymptotes include a right angle is said to I rectangular hyperbola or we can say that, if the lengths of transver: and conjugate axes of any hyperbola be equal, then it is said to be rectangular hyperbola.

        i.e., In a hyperbola  . if b = a, then it said to be rectangular hyperbola. The eccentricity of a rectangular hyperbola is always √2.

        Rectangular Hyperbola of the Form x2 – y2 = a2

          1. Asymptotes are perpendicular lines i.e., x ± y = 0
          2. Eccentricity e = √2.
          3. Centre (0, 0)

        4. Foci (± -√2 a, 0)

        1. Vertices A(a, 0) and A1 (—a, 0)
        2. Directrices x = + a/√2
        3. Latusrectum = 2a
        4. Parametric form x = a sec θ, y = a tan θ
        5. Equation of tangent, x sec θ – y tan θ = a

        Rectangular Hyperbola of the Form xy = c2

        1. Asymptotes are perpendicular lines i.e., x = 0 and y = 0
        2. Eccentricity e = √2
        3. Centre (0, 0)

        4. Foci S(√2c, √2c), S1(-√2c, -√2c)

        1. Vertices A(c, c), A1(— c,— c) 
        2. Directrices x + y = ±√2c
        3. Latusrectum = 2√2c
        4. Parametric form x = ct, y = c/t

        Tangent Equation of Rectangular Hyperbola xy = c2

        1. Point Form The equation of tangent at (x1, y1) to the rectangular hyperbola is xy1 + yx1 = 2c2 or (x/x1 + y/y1) = 2.
        2. Parametric Form The equation of tangent at (ct, c/t) to the hyperbola is( x/t + yt) = 2c.
        3. Tangent at P(ct1, c/t1) and Q (ct2, c/t2) to the rectangular hyperbola intersect a 
        4. The equation of the chord of contact of tangents drawn from a point (x1, y1) to the rectangular hyperbola is xy1 + yx1 = 2c2.

         

        Normal Equation of Rectangular Hyperbola xy = c2

        1. Point Form The equation of the normal at (x1, y1) to the rectangular hyperbola is xx1 – yy1 = x12 – y12.
        2. Parametric Form The equation of the normal at ( ct, c/t)to the rectangular hyperbola xy

        = c2 is xt3 — yt — ct4 + c = O.

        1. The equation of the normal at( ct, c/t)is a fourth degree equation t in t. So, in general four normals can be drawn from a point to the hyperbola xy = c2.

        Important Points to be Remembered

        1.  The point (x1, y1) lies outside, on or inside the hyperbola according as 
        2. The combined equation of the pairs of tangent drawn from a point P(x1, y1) lying outside the hyperbola 
        3. The equation of the chord of the hyperbola xy = c2 whose mid-point is (x1, y1) is xy1 + yx1 = 2x1y1

        or t = S1

        1. Equation of the chord joining t1, t2 on xy = t2 is x + yt1t2 = c(t1 + t2)
        2. Eccentricity of the rectangular hyperbola is √2 and the angle between asymptotes is 90°.
        3. If a triangle is inscribed in a rectangular hyperbola, then its orthocentre lies on the hyperbola.
        4. Any straight line parallel to an asymptotes of a hyperbola intersects the
        Prev Chapter Notes – Conic Sections Ellipse
        Next Introduction to Rectangular Cartesian Coordinate Geometry (2D), Distance b/w two points

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